You start with the number 1. You can create a new number by applying an operation on two existing numbers (can be the same). The operations are +, - and *. What is the fewest number of steps needed to reach the number 2020? Bonus question: can you find multiple solutions?

$\begingroup$"You start with the number 1. You can create a new number by applying an operation on two existing numbers (can be the same)." - We're given 1. Where would these "existing numbers" come from?$\endgroup$
– Sam AxeDec 18 '19 at 16:46

1

$\begingroup$@SamAxe My question exactly. I am not even starting the puzzle, just trying to guess what the other number "should" be.$\endgroup$
– GnudiffDec 18 '19 at 19:01

Outputs for n < 7:
A total of 0 solutions were found for 1 operations
A total of 0 solutions were found for 2 operations
A total of 0 solutions were found for 3 operations
A total of 0 solutions were found for 4 operations
A total of 0 solutions were found for 5 operations
A total of 0 solutions were found for 6 operations

$\begingroup$Thanks! 6 and 11 are essentially the same, the difference is the second operation is addition in 6 and multiplication in 11. Whether or not that difference is trivial would reduce down some of the other solutions as well.$\endgroup$
– Joe HabelDec 19 '19 at 6:43

$\begingroup$I like how your original solution ends by multiplying the two factors of 2020 that when summed together form the lowest number. This was my strategy but you beat me to it!$\endgroup$
– RorxorDec 18 '19 at 4:15

PARTIAL. Following the idea of @Engineer Toast, let us concentrate on the optimality part (which is actually the essence of the question: see "fewest").

The lower bound is 6. It's easy to see that the highest obtainable numbers are in decreasing order: 256,81,64,36 after the 4th step. We can't use addition as the 5th operation, nor multiplying since 2020 is not divisble by these numbers, and 36 should be multiplyed again with a higher number than itself. So the remaining question: is 6 operations possible, or not.